2025/06/26 by Takuya Koriyama, Koriyama, Takuya
Mathematics · Computer Science · #Markov Chains and Monte Carlo Methods #Bayesian Methods and Mixture Models
paper · pdf · doi:10.48550/arxiv.2506.21527
We study the asymptotic properties of parameter estimation and predictive inference under the exchangeable Gibbs partition, characterized by a discount parameter α∈(0,1) and a triangular array vn,k satisfying a backward recursion. Assuming that vn,k admits a mixture representation over the Ewens--Pitman family (α, θ), with θ integrated by an unknown mixing distribution, we show that the (quasi) maximum likelihood estimator αn (QMLE) for α is asymptotically mixed normal. This generalizes earlier results for the Ewens--Pitman model to a more general class. We further study the predictive task of estimating the probability simplex pn, which governs the allocation of the (n+1)-th item, conditional on the current partition of [n]. Based on the asymptotics of the QMLE αn, we construct an estimator pn and derive the limit distributions of the f-divergence Df(pn||pn) for general convex functions f, including explicit results for the TV distance and KL divergence. These results lead to asymptotically valid confidence intervals for both parameter estimation and prediction.